
By the strengthened principle of positivity of mass (Theorem REF ), for any \(\mu ,\nu \in \mathfrak {M}^+\)  such that \(\kappa _\alpha \mu \leqslant \kappa _\alpha \nu \)  n.e. on \(F_1\) , we have \(\mu (\mathbb {R}^n)\leqslant \nu (\mathbb {R}^n)\) , the set \(F_1\)  being not 2-thin at infinity [1]}. On the other hand, the set \(F_2\)  is 2-thin at infinity [1]}, and hence, according to Theorem REF , there are \(\mu _0,\nu _0\in \mathfrak {M}^+\)  such that
\(\nu _0(\mathbb {R}^n)<\mu _0(\mathbb {R}^n),\text{ \ though \ }\kappa _\alpha \mu _0=\kappa _\alpha \nu _0\text{\ n.e.\ on $F_2$}.\) 
